The equivalent is:

python3 -m http.server

Using 2to3 utility.

$ cat try.py
import SimpleHTTPServer

$ 2to3 try.py
RefactoringTool: Skipping implicit fixer: buffer
RefactoringTool: Skipping implicit fixer: idioms
RefactoringTool: Skipping implicit fixer: set_literal
RefactoringTool: Skipping implicit fixer: ws_comma
RefactoringTool: Refactored try.py
--- try.py  (original)
+++ try.py  (refactored)
@@ -1 +1 @@
-import SimpleHTTPServer
+import http.server
RefactoringTool: Files that need to be modified:
RefactoringTool: try.py

Like many *nix utils, 2to3 accepts stdin if the argument passed is -. Therefore, you can test without creating any files like so:

$ 2to3 - <<< "import SimpleHTTPServer"

In addition to Petr’s answer, if you want to bind to a specific interface instead of all the interfaces you can use -b or --bind flag.

python -m http.server 8000 --bind 127.0.0.1

The above snippet should do the trick. 8000 is the port number. 80 is used as the standard port for HTTP communications.

In one of my projects I run tests against Python 2 and 3. For that I wrote a small script which starts a local server independently:

$ python -m $(python -c 'import sys; print("http.server" if sys.version_info[:2] > (2,7) else "SimpleHTTPServer")')
Serving HTTP on 0.0.0.0 port 8000 ...

As an alias:

$ alias serve="python -m $(python -c 'import sys; print("http.server" if sys.version_info[:2] > (2,7) else "SimpleHTTPServer")')"
$ serve
Serving HTTP on 0.0.0.0 port 8000 ...

Please note that I control my Python version via conda environments, because of that I can use python instead of python3 for using Python 3.

As everyone has mentioned http.server module is equivalent to python -m SimpleHTTPServer.

But as a warning from https://docs.python.org/3/library/http.server.html#module-http.server

Warning: http.server is not recommended for production. It only implements basic security checks.

Usage

http.server can also be invoked directly using the -m switch of the interpreter.

python -m http.server

The above command will run a server by default on port number 8000. You can also give the port number explicitly while running the server

python -m http.server 9000

The above command will run an HTTP server on port 9000 instead of 8000.

By default, server binds itself to all interfaces. The option
-b/–bind specifies a specific address to which it should bind. Both IPv4 and IPv6 addresses are supported. For example, the following
command causes the server to bind to localhost only:

python -m http.server 8000 --bind 127.0.0.1

or

python -m http.server 8000 -b 127.0.0.1

Python 3.8 version also supports IPv6 in the bind argument.

Directory Binding

By default, server uses the current directory. The option -d/--directory specifies a directory to which it should serve the files. For example, the following command uses a specific directory:

python -m http.server --directory /tmp/

Directory binding is introduced in python 3.7

Just wanted to add what worked for me:
python3 -m http.server 8000 (you can use any port number here except the ones which are currently in use)